Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{6z^2 + 18z - 240}{7z^2 + 126z + 560}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {6(z^2 + 3z - 40)} {7(z^2 + 18z + 80)} $ $ y = \dfrac{6}{7} \cdot \dfrac{z^2 + 3z - 40}{z^2 + 18z + 80} $ Next factor the numerator and denominator. $ y = \dfrac{6}{7} \cdot \dfrac{(z + 8)(z - 5)}{(z + 8)(z + 10)}$ Assuming $z \neq -8$ , we can cancel the $z + 8$ $ y = \dfrac{6}{7} \cdot \dfrac{z - 5}{z + 10}$ Therefore: $ y = \dfrac{ 6(z - 5)}{ 7(z + 10)}$, $z \neq -8$